Nonlinear Dynamics 34: 293–307, 2003.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.Centrifugal Pendulum Vibration Absorbers: An Experimentaland Theoretical InvestigationALAN G. HADDOW and STEVEN W. SHAWDepartment of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, U.S.A.(Received: 19 June 2001; accepted: 21 July 2002)Abstract. This paper presents results from tests completed on a rotor system fitted with pendulum-type centrifugaltorsional vibration absorbers. A review of the associated theoretical background is also given and the experimentaland theoretical results are compared and contrasted. An overview of the test apparatus is provided and its uniquefeatures are discussed. To the best knowledge of the authors, this is the first time that a systematic study of thedynamic behavior of torsional vibration absorbers has been undertaken in a controlled environment.Keyw ords: Absorbers, CPVA, TVA, torsional.1. IntroductionCentrifugal Pendulum Vibration Absorbers (CPVAs) are used to suppress torsional vibrationsin rotating machinery. They are widely employed in light aircraft engines and helicopterrotors. They are also used in some high-performance automotive racing engines, diesel cam-shafts, and other places where torsional vibrations at a given order are troublesome. Theseabsorbers consist essentially of pendulums suspended from a rotor such that they can oscillatein a plane perpendicular to the axis of rotation. By proper selection of the suspension pointand effective pendulum length, the absorbers can be tuned to address torsional vibrations at agiven order of rotation. A key characteristic of these absorbers is that they remain tuned at allrotation speeds.In application, these absorber systems are composed of a set of identical pendulums thatare stationed along and about the axis of rotation. Recent analytical work has shown thatsystems of identical absorbers are dynamically very rich and can exhibit a variety of dynamicinstabilities that affect their operating range and effectiveness [1].While pendulum absorbers have been in use for over half a century, they continue to findnew applications due to increasingly stringent vibration requirements in a variety of fields.Previous research has focused mainly on analysis and simulations for systematic design stud-ies, but experimental studies are very limited and have all been tailored to specific applications[2]. In contrast, the present experimental study is, to the authors’ knowledge, the first system-atic exploration of a design space for pendulum absorbers. Specifically, it is the first studyin which one can adjust the relative tuning of the absorber to the applied torque, as well asthe torque amplitude, ‘on the fly’. This allows one to quickly determine the optimal absorbertuning for maximum effectiveness over a given torque range.The paper is organized as follows: The next section provides a review of the associatedtheoretical background. Governing equations, their approximate solutions, and definitionsof the various parameters are stated without detailed derivation, but relevant references are294 A.G. Haddow and S. W. ShawF igure 1. A schematic view of CPVAs mounted on a rotor of inertia J . For clarity, only the ithabsorber, ofindividual mass mi,isshown.Siis its displacement measured along the arc length of the path.cited. The various steady-state solutions that are possible and their stability are also discussed.An overview of the experimental facility and its capabilities are described, and experimentalresults are presented and compared to the theoretical predictions. The paper closes with someconcluding remarks and discusses the direction of future work.2. Theoretical BackgroundThis section outlines the analysis that provides the theoretical predictions used in the compar-isons with experimental data. Only a summary is provided; the full development can be foundin [1].2.1. EQUATIONS OF MOTIONThe simplest model that captures the essential dynamics of interest consists of a rigid rotor towhich are fitted several point mass absorbers that move along prescribed paths relative to therotor. The rotor is subjected to an applied torque that fluctuates at a given order of rotation, n,and the absorber paths are selected such that their motion counteracts the applied torque.We consider a system of N CPVAs mounted on a rotor of inertia J (Figure 1). The equa-tions of motion for the absorbers and the rotor are derived using Lagrange’s method for generalabsorber paths. It is convenient to recast these equations using the rotor angle θ as the inde-pendent variable, replacing time. This renders the excitation in a more standard form in whichit is periodic in the independent variable. This change of variable, nondimensionalization ofthe system parameters and dynamic variables, and the assumption that all absorber massesand paths are identical, results in the following equations of motion [1]vsi+[si+˜g(si)]v−12dx(si)dsv =−µasi,i= 1,...,N, (1)εNNi=1dx(si)dssiv2+ x(si)vv+˜g(si)sivv+˜g(si)siv2+d ˜g(si)dss2iv2 + vv=εNµaNi=1˜g(si)siv − µov + o+ (θ), (2)Centrifugal Pendulum Vibration Absorbers 295where terms are defined as follows: The generalized coordinates for the N + 1 degrees offreedom are: sirepresents the displacement of the ithabsorber, as measured along the arclength of the path and normalized by Ro(defined below), and v is the nondimensional measureof the rotor angular velocity, normalized by the average rotor angular velocity . Primesdenote differentiation with respect to θ. The absorber path is described in the functions x(s),which represents the square of the distance from the center of rotation to the absorber at itslocation s,and˜g(s) =x(s) −14dx(s)ds 2,which is related to the tangent of the path. The nondimensional damping coefficients for theabsorbers and rotor are denoted by µa,andµo, respectively. The parameter ε represents theratio of the ef fective rotational inertia of all absorbers to J . For a point mass ε = (mR2o)/Jwhere m is the total absorber mass and Rois the distance from the center of rotation to theabsorber when it is at its vertex, corresponding to s = 0. The torque is composed of twoterms, a constant, o, and a fluctuating component, (θ). These torques have been normalizedby the kinetic energy of the rotor, J2. The fluctuating torque generally contains